(*

This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2005, 2006 Slawomir Kolodynski

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*)

header{*\isaheader{Field\_ZF.thy}*}

theory Field_ZF imports Ring_ZF

begin

text{*This theory covers basic facts about fields.*}

section{*Definition and basic properties*}

text{*In this section we define what is a field and list the basic properties

of fields. *}

text{*Field is a notrivial commutative ring such that all

non-zero elements have an inverse. We define the notion of being a field as

a statement about three sets. The first set, denoted @{text "K"} is the

carrier of the field. The second set, denoted @{text "A"} represents the

additive operation on @{text "K"} (recall that in ZF set theory functions

are sets). The third set @{text "M"} represents the multiplicative operation

on @{text "K"}.*}

definition

"IsAfield(K,A,M) ≡

(IsAring(K,A,M) ∧ (M {is commutative on} K) ∧

TheNeutralElement(K,A) ≠ TheNeutralElement(K,M) ∧

(∀a∈K. a≠TheNeutralElement(K,A)-->

(∃b∈K. M`⟨a,b⟩ = TheNeutralElement(K,M))))"

text{*The @{text "field0"} context extends the @{text "ring0"}

context adding field-related assumptions and notation related to the

multiplicative inverse. *}

locale field0 = ring0 K A M for K A M +

assumes mult_commute: "M {is commutative on} K"

assumes not_triv: "\<zero> ≠ \<one>"

assumes inv_exists: "∀a∈K. a≠\<zero> --> (∃b∈K. a·b = \<one>)"

fixes non_zero ("K⇩_{0}")

defines non_zero_def[simp]: "K⇩_{0}≡ K-{\<zero>}"

fixes inv ("_¯ " [96] 97)

defines inv_def[simp]: "a¯ ≡ GroupInv(K⇩_{0},restrict(M,K⇩_{0}×K⇩_{0}))`(a)";

text{*The next lemma assures us that we are talking fields

in the @{text "field0"} context.*}

lemma (in field0) Field_ZF_1_L1: shows "IsAfield(K,A,M)"

using ringAssum mult_commute not_triv inv_exists IsAfield_def

by simp;

text{*We can use theorems proven in the @{text "field0"} context whenever we

talk about a field.*}

lemma field_field0: assumes "IsAfield(K,A,M)"

shows "field0(K,A,M)"

using assms IsAfield_def field0_axioms.intro ring0_def field0_def

by simp;

text{*Let's have an explicit statement that the multiplication

in fields is commutative.*}

lemma (in field0) field_mult_comm: assumes "a∈K" "b∈K"

shows "a·b = b·a"

using mult_commute assms IsCommutative_def by simp;

text{*Fields do not have zero divisors.*}

lemma (in field0) field_has_no_zero_divs: shows "HasNoZeroDivs(K,A,M)"

proof -

{ fix a b assume A1: "a∈K" "b∈K" and A2: "a·b = \<zero>" and A3: "b≠\<zero>"

from inv_exists A1 A3 obtain c where I: "c∈K" and II: "b·c = \<one>"

by auto;

from A2 have "a·b·c = \<zero>·c" by simp;

with A1 I have "a·(b·c) = \<zero>"

using Ring_ZF_1_L11 Ring_ZF_1_L6 by simp

with A1 II have "a=\<zero> "using Ring_ZF_1_L3 by simp }

then have "∀a∈K.∀b∈K. a·b = \<zero> --> a=\<zero> ∨ b=\<zero>" by auto;

then show ?thesis using HasNoZeroDivs_def by auto;

qed;

text{*$K_0$ (the set of nonzero field elements is closed with respect

to multiplication.*}

lemma (in field0) Field_ZF_1_L2:

shows "K⇩_{0}{is closed under} M"

using Ring_ZF_1_L4 field_has_no_zero_divs Ring_ZF_1_L12

IsOpClosed_def by auto;

text{*Any nonzero element has a right inverse that is nonzero.*}

lemma (in field0) Field_ZF_1_L3: assumes A1: "a∈K⇩_{0}"

shows "∃b∈K⇩_{0}. a·b = \<one>"

proof -

from inv_exists A1 obtain b where "b∈K" and "a·b = \<one>"

by auto;

with not_triv A1 show "∃b∈K⇩_{0}. a·b = \<one>"

using Ring_ZF_1_L6 by auto;

qed;

text{*If we remove zero, the field with multiplication

becomes a group and we can use all theorems proven in

@{text "group0"} context.*}

theorem (in field0) Field_ZF_1_L4: shows

"IsAgroup(K⇩_{0},restrict(M,K⇩_{0}×K⇩_{0}))"

"group0(K⇩_{0},restrict(M,K⇩_{0}×K⇩_{0}))"

"\<one> = TheNeutralElement(K⇩_{0},restrict(M,K⇩_{0}×K⇩_{0}))"

proof-

let ?f = "restrict(M,K⇩_{0}×K⇩_{0})"

have

"M {is associative on} K"

"K⇩_{0}⊆ K" "K⇩_{0}{is closed under} M"

using Field_ZF_1_L1 IsAfield_def IsAring_def IsAgroup_def

IsAmonoid_def Field_ZF_1_L2 by auto;

then have "?f {is associative on} K⇩_{0}"

using func_ZF_4_L3 by simp;

moreover

from not_triv have

I: "\<one>∈K⇩_{0}∧ (∀a∈K⇩_{0}. ?f`⟨\<one>,a⟩ = a ∧ ?f`⟨a,\<one>⟩ = a)"

using Ring_ZF_1_L2 Ring_ZF_1_L3 by auto;

then have "∃n∈K⇩_{0}. ∀a∈K⇩_{0}. ?f`⟨n,a⟩ = a ∧ ?f`⟨a,n⟩ = a"

by blast;

ultimately have II: "IsAmonoid(K⇩_{0},?f)" using IsAmonoid_def

by simp;

then have "monoid0(K⇩_{0},?f)" using monoid0_def by simp;

moreover note I

ultimately show "\<one> = TheNeutralElement(K⇩_{0},?f)"

by (rule monoid0.group0_1_L4);

then have "∀a∈K⇩_{0}.∃b∈K⇩_{0}. ?f`⟨a,b⟩ = TheNeutralElement(K⇩_{0},?f)"

using Field_ZF_1_L3 by auto;

with II show "IsAgroup(K⇩_{0},?f)" by (rule definition_of_group)

then show "group0(K⇩_{0},?f)" using group0_def by simp

qed;

text{*The inverse of a nonzero field element is nonzero.*}

lemma (in field0) Field_ZF_1_L5: assumes A1: "a∈K" "a≠\<zero>"

shows "a¯ ∈ K⇩_{0}" "(a¯)⇧^{2}∈ K⇩_{0}" "a¯ ∈ K" "a¯ ≠ \<zero>"

proof -

from A1 have "a ∈ K⇩_{0}" by simp;

then show "a¯ ∈ K⇩_{0}" using Field_ZF_1_L4 group0.inverse_in_group

by auto;

then show "(a¯)⇧^{2}∈ K⇩_{0}" "a¯ ∈ K" "a¯ ≠ \<zero>"

using Field_ZF_1_L2 IsOpClosed_def by auto

qed;

text{*The inverse is really the inverse.*}

lemma (in field0) Field_ZF_1_L6: assumes A1: "a∈K" "a≠\<zero>"

shows "a·a¯ = \<one>" "a¯·a = \<one>"

proof -

let ?f = "restrict(M,K⇩_{0}×K⇩_{0})"

from A1 have

"group0(K⇩_{0},?f)"

"a ∈ K⇩_{0}"

using Field_ZF_1_L4 by auto;

then have

"?f`⟨a,GroupInv(K⇩_{0}, ?f)`(a)⟩ = TheNeutralElement(K⇩_{0},?f) ∧

?f`⟨GroupInv(K⇩_{0},?f)`(a),a⟩ = TheNeutralElement(K⇩_{0}, ?f)"

by (rule group0.group0_2_L6);

with A1 show "a·a¯ = \<one>" "a¯·a = \<one>"

using Field_ZF_1_L5 Field_ZF_1_L4 by auto;

qed;

text{*A lemma with two field elements and cancelling.*}

lemma (in field0) Field_ZF_1_L7: assumes "a∈K" "b∈K" "b≠\<zero>"

shows

"a·b·b¯ = a"

"a·b¯·b = a"

using assms Field_ZF_1_L5 Ring_ZF_1_L11 Field_ZF_1_L6 Ring_ZF_1_L3

by auto;

section{*Equations and identities*}

text{*This section deals with more specialized identities that are true in

fields.*}

text{*$a/(a^2) = 1/a$.*}

lemma (in field0) Field_ZF_2_L1: assumes A1: "a∈K" "a≠\<zero>"

shows "a·(a¯)⇧^{2}= a¯"

proof -

have "a·(a¯)⇧^{2}= a·(a¯·a¯)" by simp;

also from A1 have "… = (a·a¯)·a¯"

using Field_ZF_1_L5 Ring_ZF_1_L11

by simp;

also from A1 have "… = a¯"

using Field_ZF_1_L6 Field_ZF_1_L5 Ring_ZF_1_L3

by simp;

finally show "a·(a¯)⇧^{2}= a¯" by simp;

qed;

text{*If we multiply two different numbers by a nonzero number, the results

will be different.*}

lemma (in field0) Field_ZF_2_L2:

assumes "a∈K" "b∈K" "c∈K" "a≠b" "c≠\<zero>"

shows "a·c¯ ≠ b·c¯"

using assms field_has_no_zero_divs Field_ZF_1_L5 Ring_ZF_1_L12B

by simp;

text{*We can put a nonzero factor on the other side of non-identity

(is this the best way to call it?) changing it to the inverse.*}

lemma (in field0) Field_ZF_2_L3:

assumes A1: "a∈K" "b∈K" "b≠\<zero>" "c∈K" and A2: "a·b ≠ c"

shows "a ≠ c·b¯"

proof -

from A1 A2 have "a·b·b¯ ≠ c·b¯"

using Ring_ZF_1_L4 Field_ZF_2_L2 by simp;

with A1 show "a ≠ c·b¯" using Field_ZF_1_L7

by simp;

qed;

text{*If if the inverse of $b$ is different than $a$, then the

inverse of $a$ is different than $b$.*}

lemma (in field0) Field_ZF_2_L4:

assumes "a∈K" "a≠\<zero>" and "b¯ ≠ a"

shows "a¯ ≠ b"

using assms Field_ZF_1_L4 group0.group0_2_L11B

by simp;

text{*An identity with two field elements, one and an inverse.*}

lemma (in field0) Field_ZF_2_L5:

assumes "a∈K" "b∈K" "b≠\<zero>"

shows "(\<one> \<ra> a·b)·b¯ = a \<ra> b¯"

using assms Ring_ZF_1_L4 Field_ZF_1_L5 Ring_ZF_1_L2 ring_oper_distr

Field_ZF_1_L7 Ring_ZF_1_L3 by simp;

text{*An identity with three field elements, inverse and cancelling.*}

lemma (in field0) Field_ZF_2_L6: assumes A1: "a∈K" "b∈K" "b≠\<zero>" "c∈K"

shows "a·b·(c·b¯) = a·c"

proof -

from A1 have T: "a·b ∈ K" "b¯ ∈ K"

using Ring_ZF_1_L4 Field_ZF_1_L5 by auto;

with mult_commute A1 have "a·b·(c·b¯) = a·b·(b¯·c)"

using IsCommutative_def by simp;

moreover

from A1 T have "a·b ∈ K" "b¯ ∈ K" "c∈K"

by auto;

then have "a·b·b¯·c = a·b·(b¯·c)"

by (rule Ring_ZF_1_L11);

ultimately have "a·b·(c·b¯) = a·b·b¯·c" by simp;

with A1 show "a·b·(c·b¯) = a·c"

using Field_ZF_1_L7 by simp;

qed

section{*1/0=0*}

text{* In ZF if $f: X\rightarrow Y$ and $x\notin X$ we have $f(x)=\emptyset$.

Since $\emptyset$ (the empty set) in ZF is the same as zero of natural numbers we

can claim that $1/0=0$ in certain sense. In this section we prove a theorem that

makes makes it explicit.*}

text{*The next locale extends the @{text "field0"} locale to introduce notation

for division operation.*}

locale fieldd = field0 +

fixes division

defines division_def[simp]: "division ≡ {⟨p,fst(p)·snd(p)¯⟩. p∈K×K⇩_{0}}"

fixes fdiv (infixl "\<fd>" 95)

defines fdiv_def[simp]: "x\<fd>y ≡ division`⟨x,y⟩"

text{*Division is a function on $K\times K_0$ with values in $K$.*}

lemma (in fieldd) div_fun: shows "division: K×K⇩_{0}-> K"

proof -

have "∀p ∈ K×K⇩_{0}. fst(p)·snd(p)¯ ∈ K"

proof

fix p assume "p ∈ K×K⇩_{0}"

hence "fst(p) ∈ K" and "snd(p) ∈ K⇩_{0}" by auto

then show "fst(p)·snd(p)¯ ∈ K" using Ring_ZF_1_L4 Field_ZF_1_L5 by auto

qed

then have "{⟨p,fst(p)·snd(p)¯⟩. p∈K×K⇩_{0}}: K×K⇩_{0}-> K"

by (rule ZF_fun_from_total)

thus ?thesis by simp

qed

text{*So, really $1/0=0$. The essential lemma is @{text "apply_0"} from standard

Isabelle's @{text "func.thy"}.*}

theorem (in fieldd) one_over_zero: shows "\<one>\<fd>\<zero> = 0"

proof-

have "domain(division) = K×K⇩_{0}" using div_fun func1_1_L1

by simp

hence "⟨\<one>,\<zero>⟩ ∉ domain(division)" by auto

then show ?thesis using apply_0 by simp

qed

end